On the higher order mode coupler design for damped accelerating structures

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On the higher order mode coupler design for damped

accelerating structures

J. Gao

To cite this version:

On the Higher Order Mode Coupler

Design for Damped Accelerating Structures

Jie Gao

Laboratoire de L'AccÂ

elÂ

erateur LinÂeaire

IN2P3-CNRS et UniversitÂ

e de Paris-Sud

91405 Orsay cedex, France

Abstract

In this paper we will discuss mainly damped structures and class them into three types. The procedures to make a higher order mode (HOM) coupler design have been given, and some useful analytical formulae, which can be used to determine the coupling apertures' dimensions and the equivalent loaded quality factors, have been derived. It is shown that few couplers to dampe a de-tuned structure is not at all ef®cient.

I. Introduction

For the linear colliders working in the multibunch mode the long range wake®elds have to be properly controlled by using detuned, damped or damped detuned structures depending on the concrete machine design parameters. In this paper we try to class damped structures into three types and to make a detail discus-sion about how to determine the coupling aperture dimendiscus-sion of a HOM coupler and how to estimate its effect on the damped HOM. In section 2, the ®rst type damped structure is discussed, where each cavity is damped by HOM couplers. In section 3, the second type damped structure is de®ned as a section of con-stant impedance structure is damped by a HOM coupler. The es-sential difference between the ®rst and the second type damped structures is demonstrated. To estimate the effect of this HOM coupler an equivalent quality factor has been introduced and for-mularized. Finally, in section 4, damped detuned structures are discussed and it is shown that few couplers to dampe a detuned structure is not at all ef®cient.

II. The First Type of Damped Structure

We consider a single pill-box resonant rf cavity resonating at a resonant mode, for example the TM110mode. The quality factor

corresponding to this mode is

Q

110If this cavity is loaded with

some waveguides, the loaded quality factor

Q

L;

110is de®ned as

Q

L;

110 =

Q

0

;

110 1+



N;

110 (1)

where



N;

110is called the coupling coef®cient between

waveg-uides and the resonant cavity corresponding to the TM110mode.

Where the subscript

N

denotes the number of the waveguides. If the cavity is loaded with four waveguides distributed uniformly around the cavity's cylindrical surf ace, from the general expres-sion derived in refs. [1] and [2] one ®nds that



4

;

110is TM110

mode polarization independent and can be expressed analytically

D h d

2a 2R

Figure. 1. Constant impedance accelerating structure as



4

;

110 (

l

)=

Z

0

kk

10

l

6

e

2



c

t

J

0 1 (

u

11 ) 2 144(

ln

( 4

lw

) 1) 2

ABRR

s;

110 (

R

+

h

)

J

2 2 (

u

11 ) (2) where

k

= 2

=

110,

k

10 =

k

(1 (



110

=

2

A

) 2 ) 1

=

2 ,



c

= (2

=

110 )((



110

=

2

l

) 2 1) 1

=

2 ,

R

s;

110 =(

c

0

=

110 ) 1

=

2 ,



0

is the magnetic permeability and



is the electric conductivity,

A

and

B

are the width and the height of HOM waveguides,

l

and

w

are the width and the height of the four rectangular coupling slots with

l

parallel to the magnetic ®eld, and

t

is the wall thick-ness between cavity inner surface and waveguide. Now, if many of this kind of waveguide loaded cavities are coupled together, one obtains the ®rst kind of damped accelerating structureunder the condition that the fundamental mode is not damped. For the TM11 mode passband one can say that the loaded

Q

is almost

Q

L;

110. If

K

h

Q

L;

110

<

2there will be no coupling between

cavities for the TM11mode [3], where

K

h

is the coupling

coef-®cient for the TM11mode passband.

III. The Second Type of Damped Structure

Now we consider the second kind of damped structure which is essentially different from the ®rst. Given a constant impedance disk-loaded structure of length

L

as shown in Fig. 1, one can calculate the passband of the TM11 mode (which is

the most dangerous mode for multibunch operation). We de®ne



1

;s

as the phase shift of the TM11mode passband at which the

phase velocity equals to the velocity of light. Assuming now a charged particle is injected into the structure off axis at a velocity close to that of light, one knows that this particle will loss its en-ergy to the TM11mode mainly at synchronous phase shift



1

;s

and generates behind it so-called wake®eld

W

?

;

11

(

s

), where

s

is the distance between this exciting particle and a following test particle (of course it will generate wake®elds oscillating at other 1717

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Figure. 2.



4

;

11

=1surface with

v

g

(



1

;s

)

=c

=0

:

001

frequencies also). If0

< 

1

;s

< 

the energy of the

wake-®eld

W

?

;

11, which is distributed uniformly along the structure

length

L

just after the passage of the exciting electron, will tend to move upstream with the group velocity of

v

g

(



1

;s

). Assuming

that four distributed waveguides, for example, are connected to the ®rst cavity at the beginning of the structure, in order to absorb the energy of

W

?

;

11from the structure via the HOM waveguides

without any re¯ection, one has to match this HOM coupler to the TM11travelling wave mode working at



1

;s

by carefully

choos-ing the slot dimensions on the coupler wall (just like matchchoos-ing the fundamental mode input and output couplers of a constant impedance travelling wave accelerating structure). The coupling slots' width

l

and height

w

have to satisfy the following

rela-tion [2][4]:



4

;

11 (

l

 )=

Z

0

kk

10

l

6  exp( 2





c

t

)

J

0 1 (

u

11 ) 2 144(

ln

( 4

l



w

 ) 1) 2

ABRR

s;

110 (

R

+

h

)

J

2 2 (

u

11 )  1 (1+

Z

0

R

2

R

s;110 (

R

+

h

) (

v

g(



1;s)

c

)) =1 (3) where





c

= (2

=

110 )((



110

=

2

l

 ) 2 1) 1

=

2

and

c

is the ve-locity of light. Comparing eq. 2 with eq. 3, one ®nds out the fundamental difference between the two kinds of damped struc-ture, that is, the coupling slots of the HOM coupler of the second type damped structure should not be too large or too small but satisfy matching condition shown in eq. 3, however, for the ®rst type damped structure this limitation doesn't exist. Here we will use eq. 3 to ®nd the dimensions of HOM coupling apertures of a 3GHz accelerating structure with

a

= 1

:

5cm,

D

= 3

:

33cm,

R

= 4

:

1cm,

h

= 2

:

83cm,

A

= 4

:

4cm and

B

= 2

:

cm. Fig. 2

shows the relation between

w

and

l

for different wall thickness

at

v

g

(



1

;s

)

=c

=0

:

001.

Once the HOM coupler is matched one knows the de¯ecting force of the transverse wake®eld

W

?

;

11 on the test particle of

charge

q

which is behind the exciting particle by a distance of

s

:

F

?

;

11 =

qW

?

;

11

e

! ( 1;s ) 2Q 0;11 ( s c )

D

(

s

) (4) where

D

(

s

)=

L v

g

(



1

;s

)

s

c;

(

s



cL=V

g

(



1

;s

))

D

(

s

)=0

;

(

s > cL=v

g

(



1

;s

)) (5) where

!

(



1

;s

)is the angular frequency of the travelling TM 11

mode at the phase shift of



1

;s

,

Q

0

;

11is the quality factor of the

structure working at the angular frequency

!

(



1

;s

). It is

obvi-ous that when

v

g

(



1

;s

)=0there is no damping effect (only the

coupler cavity is damped). For those who are used to think in terms of loaded quality factor to judge the damping effect of this HOM coupler, we will use an exponential function

Le

! ( 1;s ) 2Q e;11 ( s c )

to simulate function

D

(

s

)in eq. 5 by choosing

Q

e;

11as

Q

e;

11 =

!

(



1

;s

)

L

(1

e

1 ) 2

v

g

(



1

;s

) (6) in the way that both functions drop to the the same value

Le

1

at the same

s

. Eq. 4 can be expressed as

F

?

;

11 =

qW

?

;

11

Le

! ( 1;s ) 2Q L;11 ( s c ) (7) where

Q

L;

11is called the equivalent loaded quality factor which

is

Q

L;

11 =

Q

0

;

11 1+

Q

0;11

Q

e;11 =

Q

0

;

11 1+



eq

(8) where



eq is called equivalent coupling coef®cient which has

nothing to do with the coupling coef®cients expressed in eq. 2 and eq. 3. If

Q

0

;

11



Q

e;

11,

Q

L;

11 

Q

e;

11. From eq. 6 it is

very important to note that there are two ways to decrease

Q

L;

11,

either to reduce

L

or to increase

v

g

(



1

;s

). The total number

N

c

of the HOM couplers which should be used to reach a desired

Q



L;

11for a constant impedance structure of length

L

can be

cal-culated from the following expression:

N

c

=

L!

(



1

;s

)(1

e

1 )(

Q

0

;

11

Q



L;

11 ) 2

cQ

0

;

11

Q



L;

11 (

v

g (



1;s )

c

) (9) The second kind damped structure is used, for example, in TESLA linear collider project.

IV. The Third Type of Damped Structure

The third type of damped structure is to damp a detuned struc-ture by few HOM waveguides. We will ®rst discuss it without adding HOM coupler and assuming that cavities' dimensions are different from each other adiabatically. The TM11modes

disper-sion curves of this detuned structure are shown in Fig. 3. When a particle traverse the structure off axis it will deposit part of its energy on TM11mode oscillating at the frequencies mainly from

f

1to

f

n

as shown in Fig. 3, where the subscript

n

is the total

number of different cavities in this detuned structure of length

L

. The de¯ecting force felt by a test particle can be expressed as

Figure. 3. Dispersion curve of a detuned structure

where

k

1

;i

is the TM11mode loss factor corresponding to the

ith

cavity, and it can be calculated analytically [2]

k

1

;i

=

a

2

i

u

2 11

D

2



1;s;i 4



0

hR

4

i

J

2 2 (

u

11 ) ( 2

R

i

a

i

u

11

J

1 (

u

11

R

i

a

i

)) 2 (12)





1;s;i = 2



1

;s;i

sin(



1

;s;i

h

2

D

) (13)

If we consider an uniformly detuned structure with its iris ra-dius adiabatically decreasing from

a

1at the beginning to

a

n

at

the end, the deposited energies can move inside the structure in-stead of oscillating locally. The motions of the TM11modes'

en-ergies inside the structure can be classi®ed into three different ways. For the modes deposited at the beginning of the structure they will oscillate locally since they could not propagate down-stream. For the modes generated at the end of the structure (from

f

m

to

f

n

,

v

g

(



1

;s;i

)

<

0

;i

=

m;

;n

), however, the energies

will propagate upstream and be trapped ®nally somewhere in the downstream cavities where the group velocities correspond-ing to these frequencies are equal to zero. Finally, the energies of the modes between

f

1and

f

m

will propagate upstream until

they are re¯ected by the ®rst cavity. The re¯ected energies will move downstream and ®nally be trapped in the cavities where the group velocities are zero.

To damp the TM11 modes in a detuned structure via a few

HOM couplers is the main idea of the so-called damped detuned structure currently adopted in the S-Band linear collider main linac design [5]. To demonstrate the behaviour of a HOM cou-pler let's assume that at the beginning of a detuned structure a HOM coupler (four waveguides) is installed and is matched to the

ith

TM11mode. The equivalent loaded quality factor

corre-sponding to the

ith

mode can be expressed as

Q

L;

11

;i

=

Q

0

;

11

;i

1+

Q

0;11;i 2

<v

g (



1;s;i )

>

!

(



1;s;i )

z

i (1

e

1 ) (14) where

< v

g

(



1

;s;i

)

>

is the average group velocity of the

ith

mode travelling from

z

=

z

i

to

z

 0where the HOM

cou-pler is located. The damping effects of this HOM coucou-pler on the

other modes, however, are less ef®cient and different from one to another due to mismatching. The re¯ected HOM energies by this mismatched HOM coupler will be trapped somewhere in the structure. It seems that the ®rst type damping scheme is the most suitable choice for a damped detuned structure (SLAC damped detuned structure is a good example [6]).

V. Acknowledgements

He appreciates the discussions with J. Le Duff and H. Braun.

References

[1] J. Gao, ºAnalytical formula for the coupling coef®cient



of a cavity-waveguide coupling systemº, Nucl. Instr. and

Meth., A309 (1991) p. 5.

[2] J. Gao, ºAnalytical approach and scaling laws in the design of disk-loaded travelling wave accelerating structuresº,

Par-ticle Accelerators, Vol. 43(4) (1994) pp. 235-257.

[3] J. Gao, ºThe criterion for the coupling states between cav-ities with lossesº, Nucl. Instr. and Meth., A352 (1995) pp. 661-662.

[4] J. Gao, ºAnalytical formulae for the coupling coef®cient



between a waveguide and a travelling wave structureº, Nucl.

Instr. and Meth., A330 (1993) p. 306.

[5] N. Holtkamp, ºS-band test acceleratorº, Proceedings of the LC93, Oct. 13-21 (1993) Stanford, SLAC-436, 1994. [6] N. Kroll, K. Thompson, K. Bane, R. Gluckstern, K. Ko, R.

Miller, and R. Ruth, ºHigher order mode damping for a de-tuned structureº, SLAC-PUB-6624, Aug. 1994.